A ball's height, in feet, t seconds after it is thrown is modeled by \(\mathrm{h(t) = -2t^2 + 12t -...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(t) = -2t^2 + 12t - 4}\)
  • Need slope of line connecting points at \(\mathrm{t = 1}\) and \(\mathrm{t = 4}\)
  • Answer choices: (A) -6, (B) -2, (C) 0, (D) 2

2. INFER the approach

• To find the slope of a line connecting two points, we need the slope formula
• First, we must find the coordinates of both points by evaluating \(\mathrm{h(t)}\)
• The slope formula is: \(\mathrm{slope = \frac{y_2 - y_1}{x_2 - x_1}}\)

3. SIMPLIFY by evaluating h(1)

• \(\mathrm{h(1) = -2(1)^2 + 12(1) - 4}\)
• \(\mathrm{h(1) = -2(1) + 12 - 4}\)
• \(\mathrm{h(1) = -2 + 12 - 4 = 6}\)
• First point: \(\mathrm{(1, 6)}\)

4. SIMPLIFY by evaluating h(4)

• \(\mathrm{h(4) = -2(4)^2 + 12(4) - 4}\)
• \(\mathrm{h(4) = -2(16) + 48 - 4}\)
• \(\mathrm{h(4) = -32 + 48 - 4 = 12}\)
• Second point: \(\mathrm{(4, 12)}\)

5. SIMPLIFY using the slope formula

• \(\mathrm{Slope = \frac{12 - 6}{4 - 1} = \frac{6}{3} = 2}\)

Answer: D (2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when evaluating \(\mathrm{h(4) = -2(4)^2 + 12(4) - 4}\)

Students often calculate \(\mathrm{-2(4)^2}\) as \(\mathrm{-2(4) = -8}\) instead of \(\mathrm{-2(16) = -32}\), or they make errors combining the terms \(\mathrm{-32 + 48 - 4}\). If they get \(\mathrm{h(4) = 8}\) instead of 12, their slope becomes \(\mathrm{\frac{8 - 6}{4 - 1} = \frac{2}{3}}\), which doesn't match any answer choice and leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY skill: Mixing up the slope formula or calculation order

Some students might calculate \(\mathrm{\frac{4 - 1}{12 - 6} = \frac{3}{6} = \frac{1}{2}}\) (reciprocal error) or make arithmetic mistakes in the final division. Since \(\mathrm{\frac{1}{2}}\) isn't an answer choice, this leads to random selection among the given options.

The Bottom Line:

This problem tests whether students can carefully evaluate quadratic functions and apply the slope formula. The key challenge is maintaining accuracy with negative coefficients and signed arithmetic throughout multiple calculation steps.